Fungrim entry: cf5355

\psi\!\left(z\right) = \log(z) - \frac{1}{2 z} - \sum_{n=1}^{N - 1} \frac{B_{2 n}}{2 n {z}^{2 n}} + R'_{N}(z)

Assumptions:

z \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 0}

TeX:

\psi\!\left(z\right) = \log(z) - \frac{1}{2 z} - \sum_{n=1}^{N - 1} \frac{B_{2 n}}{2 n {z}^{2 n}} + R'_{N}(z)

z \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`DigammaFunction`	$\psi\!\left(z\right)$	Digamma function
`Log`	$\log(z)$	Natural logarithm
`Sum`	$\sum_{n} f(n)$	Sum
`BernoulliB`	$B_{n}$	Bernoulli number
`Pow`	${a}^{b}$	Power
`Derivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Derivative
`StirlingSeriesRemainder`	$R_{n}\!\left(z\right)$	Remainder term in the Stirling series for the logarithmic gamma function
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("cf5355"),
    Formula(Equal(DigammaFunction(z), Add(Sub(Sub(Log(z), Div(1, Mul(2, z))), Sum(Div(BernoulliB(Mul(2, n)), Mul(Mul(2, n), Pow(z, Mul(2, n)))), For(n, 1, Sub(N, 1)))), Derivative(StirlingSeriesRemainder(N, z), For(z, z))))),
    Variables(z, N),
    Assumptions(And(Element(z, SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0))), Element(N, ZZGreaterEqual(0)))))

Topics using this entry

Digamma function